Causal Inference, Hypothesis Testing, Z-scores
POLS 3316: Statistics for Political Scientists
2023-10-28
Standard Errors - distance between sample and population data
Z-scores - probability that sample represents the true population data
+ Z- Score tablesExcept…
These are easy compared to the big issue…
That’s aspirational
Z < 1.96: “the null hypothesis is retained”
- The Theory is WrongZ < 1.96: “the null hypothesis is retained”
- The Theory is Wrong
- As writtenZ < 1.96: “the null hypothesis is retained”
- The Theory is Wrong
- As written
- In some wayPossible: “the null hypothesis is retained”
- The Theory is Wrong
- As written
- In some wayZ > 1.96:
Possible: “the null hypothesis is retained”
- The Theory is Wrong
- As written
- In some wayZ > 1.96: “the null hypothesis is rejected”
Possible: “the null hypothesis is retained”
- The Theory is Wrong
- As written
- In some wayZ > 1.96: “the null hypothesis is rejected”
- The Theory is Right??Possible: “the null hypothesis is retained”
- The Theory is Wrong
- As written
- In some wayZ > 1.96: “the null hypothesis is rejected”
- The Theory is Right??NO!!!!!!
The evidence supports the hypothesis.
The evidence supports the hypothesis.
The evidence is consistent with the theory.
The evidence supports the hypothesis.
The evidence is consistent with the theory.
The null hypothesis is rejected and the evidence is consistent with the hypothesized effect.
The evidence supports the hypothesis.
The evidence is consistent with the theory.
The null hypothesis is rejected and the evidence is consistent with the hypothesized effect.
What about certainty and proof?
Bayes Rule
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Some of the things in these notes are from courses I took, some are from assorted books, some are from these two sources which are at least somewhat readable and free:
https://egap.org/resource/10-things-to-know-about-hypothesis-testing/
https://egap.org/resource/10-things-to-know-about-causal-inference/
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Correlation \(\notequal\) causation.
+ Correlation does imply a relationship that may involve some cause and effect somewhere.
+ The relationship could go either direction
+ The relationship could involve other variables
+ Lack of correlation doesn't necessarily mean anything -- correlation is linear and causal effects aren't always linearA cause is a claim about something that didn’t happen.
+ If we say X caused Y, we mean: *If X didn't happen, Y would not happen, everything else being held the same.*The Fundamental Problem of Causal Inference
+ Our proposed cause, which did happen, is the factual
+ The thing that didn't happen is called the *counterfactual*
+ We can't actually observe the thing that didn't happen
+ The inability to observe the counterfactual is the fundamental problem of causal inference
+ Experiments are a potential way around thisCauses have to involve a possible manipulation of circumstances so that the counterfactual occurred
Statistics looks for average causal effects, not single data points or individual effects. The average effects may conflict with anecdotal evidence. This is partially because…
There can be multiple causes. (Causes are non-rival.)
Causes can be necessary, sufficient, neither, or both and still be causes.
It’s a lot easier to measure effects than to find causes.
The Null Hypothesis and counterfactuals
+ We can measure the probability an effect is due to random chance (the null hypothesis)
+ Formal hypothesis tests give us this value, the *p-value*
+ Theory provides an *alternative hypothesis* which we believe to be true based on the theory
+ Well designed hypotheses can help with the unobserved counterfactual
+ When we reject the null, we can determine that "the evidence is consistent with the alternative hypothesis" and the theoryGOVT2306, Fall 2023, Instructor: Tom Hanna